EEE 498/598 Overview of Electrical Engineering Lecture 7: Magnetostatics: Ampere’s Law Of Force; Magnetic Flux Density; Lorentz Force; Biot-savart Law; Applications Of Ampere’s Law In Integral Form; Vector Magnetic Potential; Magnetic Dipole; Magnetic Flux 1

Lecture 7 Objectives To begin our study of magnetostatics with Ampere’s law of force; magnetic flux density; Lorentz force; Biot-Savart law; applications of Ampere’s law in integral form; vector magnetic potential; magnetic dipole; and magnetic flux. 2

Overview of Electromagnetics Fundamental laws of classical electromagnetics Maxwell’s equations Special cases Electro-statics Magneto-statics Geometric Optics Electro-magnetic waves Statics: Transmission Line Theory Input from other disciplines Circuit Theory Kirchoff’s Laws 3

Magnetostatics Magnetostatics is the branch of electromagnetics dealing with the effects of electric charges in steady motion (i.e, steady current or DC). The fundamental law of magnetostatics is Ampere’s law of force. Ampere’s law of force is analogous to Coulomb’s law in electrostatics. 4

Magnetostatics (Cont’d) In magnetostatics, the magnetic field is produced by steady currents. The magnetostatic field does not allow for inductive coupling between circuits coupling between electric and magnetic fields 5

Ampere’s Law of Force Ampere’s law of force is the “law of action” between current carrying circuits. Ampere’s law of force gives the magnetic force between two current carrying circuits in an otherwise empty universe. Ampere’s law of force involves complete circuits since current must flow in closed loops. 6

Ampere’s Law of Force (Cont’d) Experimental facts: Two parallel wires carrying current in the same direction attract. Two parallel wires carrying current in the opposite directions repel.  I1 I2 F12 F21   I1 I2 F12 F21 7

Ampere’s Law of Force (Cont’d) Experimental facts: A short current-carrying wire oriented perpendicular to a long current-carrying wire experiences no force. F12 = 0  I2 I1 8

Ampere’s Law of Force (Cont’d) Experimental facts: The magnitude of the force is inversely proportional to the distance squared. The magnitude of the force is proportional to the product of the currents carried by the two wires. 9

Ampere’s Law of Force (Cont’d) The direction of the force established by the experimental facts can be mathematically represented by unit vector in direction of current I1 unit vector in direction of current I2 unit vector in direction of force on I2 due to I1 unit vector in direction of I2 from I1 10

Ampere’s Law of Force (Cont’d) The force acting on a current element I2 dl2 by a current element I1 dl1 is given by Permeability of free space m0 = 4p  10-7 F/m 11

Ampere’s Law of Force (Cont’d) The total force acting on a circuit C2 having a current I2 by a circuit C1 having current I1 is given by 12

Ampere’s Law of Force (Cont’d) The force on C1 due to C2 is equal in magnitude but opposite in direction to the force on C2 due to C1. 13

Magnetic Flux Density Ampere’s force law describes an “action at a distance” analogous to Coulomb’s law. In Coulomb’s law, it was useful to introduce the concept of an electric field to describe the interaction between the charges. In Ampere’s law, we can define an appropriate field that may be regarded as the means by which currents exert force on each other. 14

Magnetic Flux Density (Cont’d) The magnetic flux density can be introduced by writing 15

Magnetic Flux Density (Cont’d) where the magnetic flux density at the location of dl2 due to the current I1 in C1 16

Magnetic Flux Density (Cont’d) Suppose that an infinitesimal current element Idl is immersed in a region of magnetic flux density B. The current element experiences a force dF given by 17

Magnetic Flux Density (Cont’d) The total force exerted on a circuit C carrying current I that is immersed in a magnetic flux density B is given by 18

Force on a Moving Charge A moving point charge placed in a magnetic field experiences a force given by The force experienced by the point charge is in the direction into the paper. B v Q 19

Lorentz Force If a point charge is moving in a region where both electric and magnetic fields exist, then it experiences a total force given by The Lorentz force equation is useful for determining the equation of motion for electrons in electromagnetic deflection systems such as CRTs. 20

The Biot-Savart Law The Biot-Savart law gives us the B-field arising at a specified point P from a given current distribution. It is a fundamental law of magnetostatics. 21

The Biot-Savart Law (Cont’d) The contribution to the B-field at a point P from a differential current element Idl’ is given by 22

The Biot-Savart Law (Cont’d) 23

The Biot-Savart Law (Cont’d) The total magnetic flux at the point P due to the entire circuit C is given by 24

Types of Current Distributions Line current density (current) - occurs for infinitesimally thin filamentary bodies (i.e., wires of negligible diameter). Surface current density (current per unit width) - occurs when body is perfectly conducting. Volume current density (current per unit cross sectional area) - most general. 25

The Biot-Savart Law (Cont’d) For a surface distribution of current, the B-S law becomes For a volume distribution of current, the B-S law becomes 26

Ampere’s Circuital Law in Integral Form Ampere’s Circuital Law in integral form states that “the circulation of the magnetic flux density in free space is proportional to the total current through the surface bounding the path over which the circulation is computed.” 27

Ampere’s Circuital Law in Integral Form (Cont’d) By convention, dS is taken to be in the direction defined by the right-hand rule applied to dl. S dl dS Since volume current density is the most general, we can write Iencl in this way. 28

Ampere’s Law and Gauss’s Law Just as Gauss’s law follows from Coulomb’s law, so Ampere’s circuital law follows from Ampere’s force law. Just as Gauss’s law can be used to derive the electrostatic field from symmetric charge distributions, so Ampere’s law can be used to derive the magnetostatic field from symmetric current distributions. 29

Applications of Ampere’s Law Ampere’s law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution. known unknown 30

Applications of Ampere’s Law (Cont’d) In general, solutions to integral equations must be obtained using numerical techniques. However, for certain symmetric current distributions closed form solutions to Ampere’s law can be obtained. 31

Applications of Ampere’s Law (Cont’d) Closed form solution to Ampere’s law relies on our ability to construct a suitable family of Amperian paths. An Amperian path is a closed contour to which the magnetic flux density is tangential and over which equal to a constant value. 32

Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law Consider an infinite line current along the z-axis carrying current in the +z-direction: I 33

(1) Assume from symmetry and the right-hand rule the form of the field Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (1) Assume from symmetry and the right-hand rule the form of the field (2) Construct a family of Amperian paths circles of radius r where 34

Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (3) Evaluate the total current passing through the surface bounded by the Amperian path 35

Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) Amperian path 36

(4) For each Amperian path, evaluate the integral Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (4) For each Amperian path, evaluate the integral length of Amperian path. magnitude of B on Amperian path. 37

(5) Solve for B on each Amperian path Magnetic Flux Density of an Infinite Line Current Using Ampere’s Law (Cont’d) (5) Solve for B on each Amperian path 38

Applying Stokes’s Theorem to Ampere’s Law  Because the above must hold for any surface S, we must have Differential form of Ampere’s Law 39

Ampere’s Law in Differential Form Ampere’s law in differential form implies that the B-field is conservative outside of regions where current is flowing. 40

Fundamental Postulates of Magnetostatics Ampere’s law in differential form No isolated magnetic charges B is solenoidal 41

Vector Magnetic Potential Vector identity: “the divergence of the curl of any vector field is identically zero.” Corollary: “If the divergence of a vector field is identically zero, then that vector field can be written as the curl of some vector potential field.” 42

Vector Magnetic Potential (Cont’d) Since the magnetic flux density is solenoidal, it can be written as the curl of a vector field called the vector magnetic potential. 43

Vector Magnetic Potential (Cont’d) The general form of the B-S law is Note that 44

Vector Magnetic Potential (Cont’d) Furthermore, note that the del operator operates only on the unprimed coordinates so that 45

Vector Magnetic Potential (Cont’d) Hence, we have 46

Vector Magnetic Potential (Cont’d) For a surface distribution of current, the vector magnetic potential is given by For a line current, the vector magnetic potential is given by 47

Vector Magnetic Potential (Cont’d) In some cases, it is easier to evaluate the vector magnetic potential and then use B =  A, rather than to use the B-S law to directly find B. In some ways, the vector magnetic potential A is analogous to the scalar electric potential V. 48

Vector Magnetic Potential (Cont’d) In classical physics, the vector magnetic potential is viewed as an auxiliary function with no physical meaning. However, there are phenomena in quantum mechanics that suggest that the vector magnetic potential is a real (i.e., measurable) field. 49

Magnetic Dipole A magnetic dipole comprises a small current carrying loop. The point charge (charge monopole) is the simplest source of electrostatic field. The magnetic dipole is the simplest source of magnetostatic field. There is no such thing as a magnetic monopole (at least as far as classical physics is concerned). 50

Magnetic Dipole (Cont’d) The magnetic dipole is analogous to the electric dipole. Just as the electric dipole is useful in helping us to understand the behavior of dielectric materials, so the magnetic dipole is useful in helping us to understand the behavior of magnetic materials. 51

Magnetic Dipole (Cont’d) Consider a small circular loop of radius b carrying a steady current I. Assume that the wire radius has a negligible cross-section. x y b 52

Magnetic Dipole (Cont’d) The vector magnetic potential is evaluated for R >> b as 53

Magnetic Dipole (Cont’d) The magnetic flux density is evaluated for R >> b as 54

Magnetic Dipole (Cont’d) Recall electric dipole The electric field due to the electric charge dipole and the magnetic field due to the magnetic dipole are dual quantities. 55

Magnetic Dipole Moment The magnetic dipole moment can be defined as Magnitude of the dipole moment is the product of the current and the area of the loop. Direction of the dipole moment is determined by the direction of current using the right-hand rule. 56

Magnetic Dipole Moment (Cont’d) We can write the vector magnetic potential in terms of the magnetic dipole moment as We can write the B field in terms of the magnetic dipole moment as 57

Divergence of B-Field The B-field is solenoidal, i.e. the divergence of the B-field is identically equal to zero: Physically, this means that magnetic charges (monopoles) do not exist. A magnetic charge can be viewed as an isolated magnetic pole. 58

Divergence of B-Field (Cont’d) No matter how small the magnetic is divided, it always has a north pole and a south pole. The elementary source of magnetic field is a magnetic dipole. N S I N S 59

Magnetic Flux The magnetic flux crossing an open surface S is given by Wb Wb/m2 60

Magnetic Flux (Cont’d) From the divergence theorem, we have Hence, the net magnetic flux leaving any closed surface is zero. This is another manifestation of the fact that there are no magnetic charges. 61

Magnetic Flux and Vector Magnetic Potential The magnetic flux across an open surface may be evaluated in terms of the vector magnetic potential using Stokes’s theorem: 62